0580

Working with Ratios

Number

The method

  • Add the parts so the whole is expressed as a number of parts
  • One part is then the quantity divided by that total
  • Each share follows by multiplying a single part by its own number in the ratio
  • Check by adding the shares back to the original amount, which catches an arithmetic slip immediately
Exam tip

Add the parts first and divide by that total, not by the number of parts. For 240 shared as 3 : 5 the total is 8 parts, so one part is 30. Then check your answers add back to the original amount — that catches a wrong total instantly.

Worked example

Dividing a quantity in a given ratio

£540 is divided between two people in the ratio 4 : 5. Work out each share.

Solution:

  • A ratio of 4 : 5 means the money is cut into 4 + 5 = 9 equal parts, not into 4 or 5
  • Finding the value of one part is the step every ratio-sharing question turns on
  • One part = 540 ÷ 9 = £60
  • The first share takes 4 of those parts: 4 × 60 = £240
  • The second takes 5 of them: 5 × 60 = £300
  • Check by adding the shares back: 240 + 300 = 540, which is the amount divided
  • A second check is the ratio itself: 240 : 300 cancels by 60 to 4 : 5
  • Keep the shares in the order the question set them, since giving £300 to the first person is a different answer
Worked example

Dividing into three parts

96 counters are divided in the ratio 3 : 4 : 5. Work out how many counters are in each group.

Solution:

  • The method does not change when there are three parts instead of two
  • Total parts = 3 + 4 + 5 = 12
  • One part = 96 ÷ 12 = 8 counters
  • The three groups take 3, 4 and 5 of those parts
  • 3 × 8 = 24, 4 × 8 = 32 and 5 × 8 = 40
  • The groups hold 24, 32 and 40 counters
  • Check by adding: 24 + 32 + 40 = 96
  • The division came out exactly, which it must: a whole number of counters cannot be split into fractions, so a total that does not divide by the number of parts is a signal to re-read the question

When the question gives a difference instead of a total

  • The difference between two shares is worth the difference between their two numbers of parts
  • Divide to find one part, then continue exactly as before
    • In the ratio 7 : 4 one share exceeds the other by 33, and 7 − 4 = 3 parts, so one part is 11
    • The shares are 77 and 44, and the total is 11 parts, or 121

When the question gives one share instead of a total

  • Match the known quantity against the number of parts it stands for, then divide to find one part
    • Paint is mixed red to white in the ratio 5 : 2, and 45 litres of red are available
    • 5 parts is 45 litres, so one part is 9 litres, giving 18 litres of white and 63 litres of mixture

Combining two ratios

  • Two two-part ratios that share a quantity join into a single three-part ratio
  • Scale each ratio so that the shared quantity is represented by the same number in both, using the lowest common multiple of its two values
  • Write the three parts out in one line once they agree
    • With A : B = 3 : 4 and B : C = 6 : 5, the shared B is 4 and 6, whose lowest common multiple is 12
    • Scaling gives A : B = 9 : 12 and B : C = 12 : 10, so A : B : C = 9 : 12 : 10